Thermodynamic properties near the onset of loop-current order in high-$T_c$ superconducting cuprates

We have performed large-scale Monte Carlo simulations on a two-dimensional generalized Ashkin-Teller model to calculate the thermodynamic properties in the critical region near its transitions. The Ashkin-Teller model has a pair of Ising spins at each site which interact with neighboring spins through pair-wise and four-spin interactions. The model represents the interactions between orbital current loops in ${\text{CuO}}_2$ plaquettes of high-$T_c$ cuprates, which order with a staggered magnetization $M_s$ inside each unit cell in the underdoped region of the phase diagram below a temperature $T^{\ast }\left(x\right)$ which depends on doping. The pair of Ising spins per unit cell represents the directions of the currents in the links of the current loops. The generalizations are the inclusion of anisotropy in the pair-wise nearest-neighbor current-current couplings consistent with the symmetries of a square lattice and the next-nearest-neighbor pair-wise couplings. We use the Binder cumulant to estimate the correlation length exponent $\nu$ and the order-parameter exponent $\beta$. Our principal results are that in a range of parameters; the Ashkin-Teller model as well as its generalization has an order-parameter susceptibility which diverges as $T\to T^{\ast }$ and an order parameter below $T^{\ast }$. Importantly, however, there is no divergence in the specific heat. This puts the properties of the model in accord with the experimental results in the underdoped cuprates. We also calculate the magnitude of the “bump” in the specific heat in the critical region to put limits on its observability. Finally, we show that the staggered magnetization couples to the uniform magnetization $M_0$ such that the latter has a weak singularity at $T^{\ast }$ and also displays a wide critical region, also in accord with recent experiments.

Figure 1

(Color online) The circulating current phase ${\Theta }_{\text{II}}$ (Ref. 2). The Cu sites are red (large) circles while O sites are blue (small) circles. The unit cell is shown by the dashed square. A staggered magnetic-moment pattern within each unit cell that repeats from unit cell to unit cell (the curl of the directed circles) is indicated. The currents $J^x$ and $J^y$ represent the horizontal and vertical currents, respectively, to be used in the derived effective model [Eq. (1)]. Physically, they represent the coherent parts of the orbital fermionic currents in the problem.

Figure 2

(Color online) An illustration of the pseudospin $\mathbf{S}=\left(J_{\mathbf{r}}^x,J_{\mathbf{r}}^y\right)$ we used to compute the staggered order parameter and its susceptibility [Eqs. (3, 4)].

Figure 3

(Color online) Critical exponents $\alpha$, $\beta$, and $\gamma$ from the Ashkin-Teller model, as a function of the four-spin coupling ${\beta }_c{K}_4\le 0$ (Ref. 21). In this parameter range, we have $-2<\alpha \le 0$, $1/8\le \beta <1/4$, $7/4\le \gamma <7/2$, and $1\le \nu <2$.

Figure 4

(Color online) Specific heat as a function of temperature $T$ for the classical part of the model in Eq. (1), with $A=1.0$ and $K_4=0.0$, for various values of $K^{xy}=0.0,0.1,0.2,0.3$, and system size $L=512$. The amplitude of the logarithmic specific heat of the Ising model $\left(K^{xy}=0\right)$ is enhanced as $K^{xy}$ increases but the anomaly remains logarithmic. The critical temperature of the 2D pure Ising model is given by $T_c=2/\text{ln}\left(1+\sqrt{2}\right)\approx 2.27$ in units where Boltzmann constant $k_B=1$. Note also that for this set of parameters, $K^{xy}$ hardly alters $T_c$ of the model with $K^{xy}=0$.

Figure 5

(Color online) Specific heat as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.0$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$. The logarithmic specific-heat singularity of the Ising model $\left(K_4=0\right)$ is eliminated and replaced by a bump whose width increases as $\left|K_4\right|$ increases. The arrow in the lower right panel indicates $T_c$ as obtained from the peak in the susceptibility ${\chi }_s$.

Figure 6

(Color online) Specific heat as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.0$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=64,128,256,512$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$. The logarithmic specific-heat singularity of the Ising model $\left(K_4=0\right)$, is eliminated and replaced by a bump whose width increases as $\left|K_4\right|$ increases. Note how the double bump in $C_V$, which is present at smaller system sizes, disappears when $L$ is increased. When $L=512$, the results appear to be well converged.

Figure 7

(Color online) Specific heat as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=0.5$ and $K^{xy}=0.0$, for various values of $K_4=0.0,-0.1,-0.25$, and system size $L=512$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$. Compared to the case shown in Fig. 5, with $A=1.0$, precisely the same trends are seen in the evolution of the anomaly as the AT coupling $\left|K_4\right|$ is increased, only slightly more pronounced. The arrow indicates $T_c$ as obtained from the peak in the susceptibility ${\chi }_s$.

Figure 8

(Color online) Specific heat as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.1$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. The arrow indicates $T_c$ as obtained from the peak in the susceptibility ${\chi }_s$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$.

Figure 9

(Color online) Specific heat as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.2$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. The arrow indicates $T_c$ as obtained from the peak in the susceptibility ${\chi }_s$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$.

Figure 10

(Color online) Specific-heat anomaly as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.3$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$.

Figure 11

(Color online) Specific-heat anomaly as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=0.5$ and $K^{xy}=0.3$, for various values of $K_4=0.0,-0.1,-0.25$, and system size $L=512$. The vertical scale is in units of $k_B/\text{unit}\text{cell}$.

Figure 12

(Color online) The staggered order parameter, Eq. (3), as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.0$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$.

Figure 13

(Color online) The susceptibility of the staggered magnetization within each unit cell, Eq. (4), as a function of temperature $T$ for the classical part of generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.0$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. Note that the susceptibility retains the nonanalytical features of the Ising case even for parameters where the specific-heat anomaly is completely suppressed.

Figure 14

(Color online) The staggered order parameter, Eq. (3), as a function of temperature $T$ for the classical part of the generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.3$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$.

Figure 15

(Color online) The susceptibility of the staggered magnetization within each unit cell, Eq. (4), as a function of temperature $T$ for the classical part of the generalized AT model Eq. (1), with $A=1.0$ and $K^{xy}=0.3$, for various values of $K_4=0.0,-0.1,-0.25,-0.5$, and system size $L=512$. Note the marked increase in the susceptibility as $-K_4$ is increased, in contrast to the suppression of the anomaly in the specific heat.

Figure 16

(Color online) The Binder cumulant $G$, Eq. (5), as a function of $T$ for model Eq. (1), for $A=1.0$, $K^{xy}=0.1$, and $K_4=-0.25$ for various system sizes, in the absence of Ferrenberg-Swendsen reweighing. The inset shows a blowup of the temperature region where the lines for various system sizes cross, providing an estimate for $T_c$.

Figure 17

(Color online) The Binder cumulant $G$, Eq. (5), as a function of $T$ for model Eq. (1), for $A=1.0$, $K^{xy}=0.1$, and $K_4=-0.25$ for various system sizes, using Ferrenberg-Swendsen reweighing. The inset shows a blowup of the temperature region where the lines for various system sizes cross, providing an estimate for $T_c$. Note the consistency of the estimate for $T_c$ compared to Fig. 16.

Figure 18

(Color online) The Binder cumulant $G$, Eq. (5) as a function of the quantity $L^{1/\nu }\left(T-T_c\right)/T_c$, for model Eq. (1), for $A=1.0$, $K^{xy}=0.1$, and $K_4=-0.25$ for various system sizes. Ferrenberg-Swendsen reweighing of the data is used. We have taken estimates for $T_c$ from Fig. 17 and adjusted the correlation length critical exponent $\nu$ to achieve the best data collapse. As is seen, the optimal $\nu$ is extremely sensitive to the chosen value of $T_c$.

Figure 19

(Color online) The scaled staggered order parameter $L^{\beta /\nu }{M}_s$ [cf. Eq. (3)] as a function of the quantity $L^{1/\nu }\left(T-T_c\right)/T_c$, for the model Eq. (1), for $A=1.0$, $K^{xy}=0.1$, and $K_4=-0.25$ for various system sizes. Ferrenberg-Swendsen reweighing of the data is used. We have taken estimates for $T_c$ from Fig. 17 and estimates for $\nu$ from Fig. 18, and adjusted the order-parameter exponent $\beta$ to achieve the best data collapse.